The usual definition of a category states: a category $\mathbf{C}$ consists of:
- A collection $\text{ob}(\mathbf{C})$ of objects
- A collection $\text{arr}(\mathbf{C})$ of arrows
- Some rules on the behaviour of these two types of objects
Leaving aside what collection here means, I realized that I have never seen a clear definition of what an arrow in a category is.
Surely, in the typical categories like $\textbf{Set}$, $\textbf{Top}$ or $\textbf{Grp}$ arrows are functions that … But there are categories whose arrows are not "functions".
If you have a poset $(P,\le)$ then you can get a category whose objects are elements of $P$ and such that there is an arrow $x\to y$ if and only if $x\le y$ in $P$. Okay but what kind of "entity" is the arrow there?, is there a precise, rigorous definition or do just have to accept that there are objects and arrows and asume that they follow the required rules, just like we asume there some things called "numbers" that follow the rules of arithmetic?
I hope I made myself clear
Thanks!
Best Answer
There are already many excellent answers, but I want to add another perspective, already partly found in other answers, but I hope distinct enough to stand on its own.
I like to explain by analogy. Consider the question, "What is a vector?"
What is a vector?
Well, you might get you any of the following informal definitions as a response:
(I think I might know some physicists who would take issue with me calling (c) informal, but oh well).
And you might then ask, well ok, but what's a formal definition of a vector?
Let's think of some examples of vectors. The elements of $\Bbb{R}^3$, the elements of $\Bbb{R}[x]$, continuous functions from $X$ to $\Bbb{R}$, where $X$ is a topological space. These seem like fairly different objects, but the common factor here is that a vector is simply an element of a set $V$ with a specified vector space structure. I.e., the formal definition of vector is simply an element of a vector space.
Why is this useful as a definition? Well, all of the properties of vectors are already encoded in the definition of vector space. So if I tell you that $v,w$ are vectors in $V$, and $r\in\Bbb{R}$ is a scalar, then you know that $v+w$ is also a vector, and that $rv$ is a vector, and that $r(v+w)=rv+rw$. All the properties of a vector that we might find interesting are encoded in the vector space axioms.
Note also that part of this means that it's meaningless to say $v$ is a vector on its own. It's only meaningful to say that $v$ is a vector of some vector space $V$. This is good, because as an element $v$ might belong to many different vector spaces with different structures, but depending on the ambient vector space structure, $v$ might behave completely differently.
Bringing it back to arrows
Similarly, if I say $f:X\to Y$ is an arrow of a category $\mathcal{C}$. The rigorous definition of arrow here is simply that $f$ belongs to the collection of arrows $\operatorname{Arr}(\mathcal{C})$, and that the domain of $f$ is $X$ and the codomain of $f$ is $Y$. All of the other interesting properties of arrows (for example that I could compose $f:X\to Y$ with an arrow $g:Y\to Z$ to get an arrow $g\circ f:X\to Z$) are already encoded in the axioms of the category $\mathcal{C}$, and there's no need to say anything further to define arrows.
Edit:
Since comments are not permanent, I just want to edit in the link from Ethan Bolker's comment, to an excellent answer with a similar viewpoint to this one in reply to a similar (in spirit) question about "what actually is a polynomial?" The second paragraph in particular really captures what I wanted to say in my answer, (paraphrasing Ethan's answer) what really matters isn't what something actually is, but rather how it behaves.